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A three-dimensional dissipative map modeling type-IIintermittency

Françoise Argoul, A. Arneodo, P. Richetti

To cite this version:Françoise Argoul, A. Arneodo, P. Richetti. A three-dimensional dissipative map modeling type-IIintermittency. Journal de Physique, 1988, 49 (5), pp.767-775. �10.1051/jphys:01988004905076700�.�jpa-00210753�

767

A three-dimensional dissipative map modeling type-II intermittency

F. Argoul (*), A. Arneodo (**) and P. Richetti (1)

Center for Nonlinear Dynamics, Physics Department, University of Texas, Austin, TX 78712, U.S.A.(1) Centre de Recherche Paul Pascal, Domaine Universitaire, 33405 Talence Cedex, France

(Requ le 22 juillet 1987, accepte sous forme definitive le 4 janvier 1988)

Résumé. 2014 Pour décrire la transition vers le chaos par intermittence de type II, nous proposons une

application de Poincaré modèle à trois dimensions qui rend compte de l’interaction entre une bifurcation localede Hopf souscritique et une bifurcation globale homocline. Une telle description, parce qu’elle se veut dedimension minimale, affecte le degré de stabilité structurelle de ce scénario et nécessite l’adaptation duformalisme de Pomeau-Manneville au cas d’un mécanisme de réinjection homocline unidimensionnel. Nousprésentons les résultats de simulations numériques qui confirment les prédictions théoriques et ce aussi bienpour les distributions statistiques des longueurs des plages laminaires que pour la divergence en

1/f0394 des spectres de puissance au voisinage du seuil de transition. Nous discutons ensuite l’influence d’un bruitaléatoire sur le scénario d’intermittence de type II en distinguant les cas de bruits additifs et multiplicatifs.

Abstract. 2014 A three-dimensional Poincaré map model for type-II intermittency is proposed. The purpose is todescribe this transition in minimal dimension which in turn affects the degree of structural stability of thescenario. This scenario is understood in terms of the interaction of a local subcritical Hopf bifurcation and aglobal homoclinic bifurcation. The Pomeau-Manneville picture is revisited in order to account for the one-dimensional homoclinic reinjection process. The distribution of the lengths of the laminar episodes is

investigated and numerical results are shown to corroborate the theoretical predictions. The influence of noiseon type-II intermittency is discussed. The difference between additive and multiplicative noise is emphasized.

J. Phys. France 49 (1988) 767-775 MAI 1988,

Classification

Physics Abstracts05.45

1. Introduction.

The introduction by Pomeau and Manneville [1] ofthe notion of intermittency as a specific route toweak turbulence has stimulated many theoretical [2,3], numerical [4-6] and experimental [7-11] studiesduring the past few years. However, in most of thesestudies, and mainly in the experimental approaches,the emphasis has been on the analysis of intermittentregimes and not on a detailed investigation of thenature of the transition to chaos. In fact, for

intermittency to be a second-order phase transition

from ordered to disordered temporal patterns, twoingredients must actually occur simultaneously : (i) alocal instability of a limit cycle which can be either asaddle-node bifurcation (Type-I intermittency), or asubcritical Hopf bifurcation (Type-II intermittency),or a subcritical period-doubling bifurcation (Type-III intermittency) ; and (ii) a global nonlinear

mechanism, e.g. strange-attractor-like behavior,which ensures the reinjection of the dynamics in theneighborhood of the limit cycle. Then, when varyinga control parameter, a continuous transition isobserved from a periodic regime to short turbulentbursts interrupting (seemingly at random) nearlyperiodic oscillations. This legitimizes the definitionof critical exponents which characterize the evolutionof the distribution of laminar episodes. As far astype-I [12] and type-III [13] intermittencies are

concerned, it has been shown that not only the localinstability but also the reinjection mechanism

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004905076700

768

strongly condition the statistics of the laminar reg-ions. One of the issues of this paper, which is

devoted to the study of type-11 intermittency, is to

point out that the nature of the reinjection processmay even affect the degree of structural stability [14]of this scenario to chaos.

Among the three types of intermittency originallydefined by Pomeau and Manneville [1], type-IIintermittency is undoubtedly the least popular, sincefor years there have been no examples identifyingthis route in either real experiments or in simulationstudies. In fact only formal models of type-11 inter-mittency have been proposed. These models are

essentially based on the subcritical Hopf normalform [15] for mappings :

together with an artificially added reinjectionmechanism obtained by periodizing the phase spaceor by assuming a uniform random reinjection distri-bution P r (p r ) = 1/ 7T R; in a disk of radius

Rr (p r Rr ) contained in the two-dimensional un-stable manifold of the fixed point (p = 0 ). With thisworking hypothesis and according to the rotationalinvariance of (1), the reentry points are chosen atrandom with uniform probability in this disk. Gener-ally, Rr -- RQ where Re delimits the laminar episodes.For the sake of simplicity, Pomeau and Manneville[1] have considered the extreme case Rr = Re. Theprobability distribution Pf (n) of the laminar lengths(n is the number of iterates) is then predicted toscale like Pf (n) - n-2 at small laminar period n andto decay exponentially Pt - exp (- 2 en ) at large n.In addition a straightforward calculation yields :

which implies that the mean length of the laminarepisodes behaves like :

At this point let us mention that a cross-over

phenomenon is expected to occur at higher values ofE where (n) is predicted to scale like :

In a recent publication [16], we have reported onthe first numerical observation of type-11 intermitten-cy in a nonautonomous differential system. This

system is a periodically driven third-order nonlinearoscillator :

which in the absence of forcing (F = 0 ) accounts forthe interaction of a (local) subcritical Hopf bifur-cation [17-19] and a (global) Shil’nikov [20] typehomoclinic bifurcation. These conditions taken

together favor a chaotic reinjection of the dynamicsin the neighborhood of the origin and provide a nicedescription of an intermittent transition from a

steady state to a turbulent regime consisting ofchaotic bursts which from time to time emerge froma nearly stationary signal. When the periodic forcingis turned on (F =F 0), the phase space of the

dynamical system (5) enlarges from R3 to R3 x T andthis transition to chaos generalizes to type-11 inter-mittency. A steady state in the absence of forcingbecomes a periodic orbit and the intermittent signaltransforms into chaotic bursts which interrupt a

nonsaturated periodic modulation of the amplitudeof the original oscillation [16].The aim of this paper is to construct a three-

dimensional dissipative map which models the Poin-car6 map of the oscillator (5) as numerically obtainedin reference [16] when sampling the orbit in phasespace at the frequency of the external forcing. In theneighborhood of the subcritical Hopf bifurcation,this mapping will reduce to the two-dimensional

Hopf normal form (1) and only one direction willremain in which the homoclinic process can reinjectthe dynamics near the fixed point. In this sense, thethree-dimensional discrete system defined in sec-

tion 2 will provide a description of type-11 intermit-tency in minimal dimension. In our preliminarynumerical study [16] of this scenario with the differ-ential system (5), we have failed to measure accu-rately critical exponents because this estimate wouldhave required prohibitive time consuming simu-lations. In this new analysis, we will take advantageof the iterative nature of our dynamical system tocompute the statistical distribution of the laminar

episodes. In particular we will emphasize that thehomoclinic reinjection mechanism is one-dimension-al (and not two-dimensional) and that the theoreticalresults (2), (3) and (4) need to be reconsidered. Insection 3, we will revisit these predictions and pro-duce numerical outcomes which corroborate our

analytical results. Moreover we will point out thatfor this transition to be continuous one must definean experimental path which crosses the hypersurfaceof codimension 2 where both the subcritical Hopfbifurcation and the homoclinic bifurcation occur

simultaneously. In other words, type-II intermittencywill appear as a scenario whose degree of structuralstability [14] is d, = 2, unlike the original picture ofPomeau-Manneville [1] where the three types ofintermittency have a degree of structural stabilityd, =1. Furthermore, we will elaborate on the fuzzi-ness of this hypersurface which will allow us to

conclude that the degree of structural stability of thisscenario to chaos is actually d, = " 2 - E ". . Finally

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we will discuss in section 3 the robustness of type-IIintermittency to the presence of noise, consideringboth the additive and multiplicative cases.

2. A Poincare map model for type-11 intermittency :construction and simulation.

2.1 CONSTRUCTION. - In a previous work [21],one of us (A. A.) has already contributed to theconstruction of a three-dimensional Poincare mapmodel for the periodically forced oscillator (5). Thefollowing families of diffeomorphisms of R3 havebeen proposed :

which preserve the triple degeneracy (three Floquetmultipliers equal to unity) of the origin at =v = q = 0 as implemented in the unforced (F = 0)three dimensional oscillator (5). In the neighborhoodof this tricritical (hyper) surface [22], A, -v and q aresmall and we can introduce the scaled variables :

where 8 is a small positive number. Then the

diffeomorphism (6) becomes a map close to the

identity [23]

As long as 5 is not too large, we can approximate(X’ -X)/8, (Y’ - Y)15 and (Z’ - Z)/5 by de-rivatives and the mapping (8) by a system of first-order differential equations which up to leadingorder in 5, is equivalent to the unforced three-dimensional oscillator (5) with only X2 as nonlinearquadratic term :

Since such a system has been pointed out to be aparadigm for the cascade of period-doubling bifur-cations [22] (Eq. (9) is the asymptotic normal formof the 3 instability), it must be emphasized thatarbitrarily close (6 -+ 0) to the tricritical surface

J.L = v = n = 0, the three-dimensional dissipativemap (6) displays the cascade of period-doublinginvariant tori [21, 24].

Unfortunately, as discussed in reference [16], theasymptotic normal form (9) does not account for theinteraction of a subcritical Hopf bifurcation and aShil’nikov homoclinic bifurcation. This concomitantsituation requires the presence of other quadraticterms in the unforced third-order oscillator (5). Thisimplies that the Poincare map model (6) has to bemaintained at some distance from the tricritical

surface u = v = q = 0 (as characterized by smallbut finite values of 8) in order to be suited to providea local description of type-II intermittency. As notedby Arnold [25] in the study of some codimension-two bifurcations of periodic orbits, when the distanceto the polycritical condition is increased, the differ-ential system derived as just explained no longerapproximates the mapping for long times. Then oneexpects to observe some deviations from the

dynamics predicted by the continuous system (9),eventhough one takes into account higher orderterms in 5 (e.g. the missing quadratic terms). There-fore a specific analysis of the Poincare map model(6) is needed to locate the Hopf and homoclinicbifurcations. When linearizing the mapping (6)around the origin, a straightforward calculation

yields the following condition for this fixed point tobecome unstable through a Hopf bifurcation :

Let us note that equation (10) reduces to thecondition [16] JL = TJ v in the limit 6 -+ 0 along theasymptotic path (7). In the neighborhood of thecritical surface defined by equation (10), the use ofboth the center manifold theorem and the normalform techniques [15, 19, 25] allows us to reduce thePoincare map model (6) to the normal form (1) ofthe Hopf bifurcation for mappings. In equation (1),E measures the distance to the critical surface, whilethe coefficients a and b are computed on thissurface. Although the detailed expressions of thesecoefficients as functions of the parameters ki of thenonlinear terms in the mapping (6) are rather

complicated, let us mention that the arbitrariness inthe ki covers both the situations a > 0 and a 0,which correspond to subcritical and supercriticalHopf bifurcations respectively.To locate Shil’nikov (spiraling out) homoclinic

orbits in ordinary differential equations, Gaspardand Nicolis [26, 27] have developed a numericaltechnique which consists in (i) studying the intersec-tion of the two-dimensional unstable manifold of thesaddle focus with a plane which is transverse to thelocal one-dimensional stable manifold ; this intersec-tion is generated when considering several trajec-tories starting from the local unstable manifold nearthe singular point ; (ii) calculating analytically thetrace of the one-dimensional stable manifold in this

plane ; (iii) using a trial and error method to deter-

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mine the condition at which the unstable manifoldcontains the stable one. This technique can be usedto define codimension-one hypersurfaces in the

parameter space of the continuous system (9) wheresuch homoclinic bifurcations occur. Since in the limit5 H 0, the asymptotic normal form (9) provides agood approximation of the Poincare map model (6),one can adapt this numerical technique to locatehomoclinic orbits in discrete systems. Then in nearlyhomoclinic conditions, one may expect to find chaosas ensured by a theorem by Shil’nikov [20]. But onehas to be very careful because for finite values of 5,this research becomes rather tricky since one expectsthe homoclinic connection to split due to transverseintersections of the unstable and stable manifolds ofthe saddle focus. Henceforth there is no longer awell defined codimension-one homoclinic hypersur-face in the parameter space but instead a fuzzyregion around this hypersurface where this newsource of chaos may lead to hyperchaos [28] withtwo positive Lyapunov exponents : one positiveLyapunov exponent corresponding to Shil’nikovhomoclinic chaos, the second one resulting from thecreation of subsequent Smale horseshoes [29] via thetransverse intersections of the invariant manifolds ofthe fixed point. Let us mention that the width of thisfuzzy region enlarges from zero when 5 is increased.Here, our goal is to trace the homoclinic bifur-

cation i.e. the corresponding codimension " 1- e "

fuzzy region, up to a close neighborhood of thepreviously described subcritical Hopf bifurcation. Inthis way we will approximately define the codimen-sion " 2 - £" fuzzy hypersurface that our one-par-ameter numerical path will have to cross for type-IIintermittency to be observed when iterating thePoincare map model (6). In that sense, the degree ofstructural stability [14] of this scenario to chaos is

ds = " 2 - £ ". At this point, let us mention that,when increasing 5 in equation (7), higher ordernonlinear terms such that k5 X2 Z are needed in thediscrete system (6) in order to saturate the homo-clinic instability and thus to carry out the reinjectionprocess in the neighborhood of the origin avoidingthe trajectories diverging to infinity.

2.2 SIMULATION. - In the sequel of this paper, wewill describe some numerical investigations of thePoincare map model (6) following a one-parameterpath in the space of constraints defined by thestraight line :

with

goes through (at least within our numerical resol-ution) the fuzzy region previously discussed [30]. Tosimplify the notation, we will consider E = g - g *as the control parameter (in fact locally it is pro-portional to the E involved in the Hopf normal form(1)). In order to check the continuous character ofthe transition along the path (11), we have estimatedthe hysteresis range (if any) at the transition to chaosto be less than A E = 10-7.

In figure 1, we illustrate the type-II intermittencyscenario observed along the one-parameter path(11). For negative values of ê (IL u * ), the originx = y = z = 0 is asymptotically stable. When cros-sing the critical value E = 0 (u = IL *, v = v * ), thesystem loses its regularity and chaotic bursts inter-rupt episodes of laminar behavior in an apparentlyrandom fashion. Immediatly beyond the thresholdof intermittency (In (e) = - 13 ), the laminar phasesare long and only very exceptional bursts emergeoccasionally as seen in figure la. Slightly above

criticality (In (e) = - 1 ), the chaotic episodes be-come more frequent and as seen in figures Ib and lc,the oscillatory behavior of the coordinate z duringthe laminar episodes corresponds to a nonsaturatedmodulation of the amplitude of the original periodicoscillation of the periodically forced oscillator (5) as

Fig. 1. - z (in 8 5 units, with 5 = 5 x 10-2) versus thenumber n of iterates as computed with the Poincare mapmodel (6) for the parameter values given in equation (11).(a) A = 1.728 x 10-4, v = 3.087 x 10-3 ; (b)A = 1.875 x 10-4, v = 3.349 x 10-3 ; (c) is the same as

(b) but with a smaller range of n-values.

771

observed in reference [16]. When the amplitude ofthis modulation reaches some critical value, then thedynamics changes drastically and a turbulent burst isinitiated. Immediately after the chaotic intermissionthere is a reappearance of the regular behavior, thelength of which depends on the distance of the

reinjection point to the origin.In figure 2a, we illustrate a two-dimensional

projection of the dynamics onto the unstable mani-fold of the origin. Only a few iterates of the mapping(6) have been retained in order to distinguish boththe one-dimensional homoclinic reinjection mechan-ism and the very smooth spiraling behavior awayfrom the origin which is the laminar episode. Thereinjection distribution as projected onto the un-stable manifold of the origin is shown as the histog-ram in figure 2b. The whole set of reentry pointsdoes not spread out on such a surface but theyinstead fall approximately along a curve as expectedfrom a homoclinic reinjection process [16, 26, 27]. Asimilar histogram has been obtained in reference

[16] with the periodically driven third-order non-linear oscillator (5) ; it reflects the numerical findingthat the Poincare map of a homoclinic strangeattractor is a multifolded line (line x Cantor) wherethe folds are generally strongly transversally con-tracted. Therefore, this histogram shows a clearevidence that the chaotic reinjection process is

essentially one-dimensional and that consequently itbreaks the rotational invariance of the Hopf normalform (1). But this symmetry is at the heart of thePomeau-Manneville [1] analysis. Therefore we needto revise the theoretical predictions (2), (3) and (4)in order to bring them into accord with the one-dimensional deterministic reinjection mechanism inthe mapping (6).

If one assumes a white radial reentry distribution

(0 p Rr) : Pr(Pr) = 1/Rr (Or = 8 * ), then the

probability distribution Pt (n) of the laminar epi-sodes is predicted to scale like n- 3/2 at small laminarperiod n and to decay exponentially Pl(n) -exp (- 2 en ) at large n. This distribution [31] is

characterized by a mean value :

where Rr is distinguished from Rp in order to matchour numerical situation where R, is found muchsmaller than the cut-off Rl which defines the laminarepisodes. Then the scaling law (3) transforms into

while the behavior (4) at larger values of £ is

unchanged:

Figure 3 provides confirmation to these theoreticalpredictions ; the average length of the laminar

periods is plotted versus the control parameter e. Wehave considered 105 laminar periods as defined byRr = 5 x 10-2 and = 5 x 10-1. The cross-overbetween the scaling behaviors (13) and (14) is clearlyshown to occur for In (E) - - 12.A typical characteristic of intermittencies is the

1/ f ° divergencies in the small frequency limit of thespectra which reflects the arbitrarily long laminarregions observed immediately beyond criticality.Because of the one-dimensional character of thehomoclinic process [31], some of the results obtainedwith one-dimensional map models [32-34] are ex-pected to apply to type-11 intermittency as simulatedwith the three-dimensional Poincare map model (6).

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Fig. 3. - The average length of 105 laminar episodesn> versus E in log-log scales. The model parameters aregiven in equation (11). The solid line corresponds to thetheoretical prediction (13). The laminar episodes are

defined by Rr = 5 x 10-2 and Rl = 5 x 10-1.

The numerical power spectrum presented in figure 4seems to corroborate this feature : the exponentA = 0.63 ± 0.10 extracted from a mean square fit ofthe power spectrum vs. the frequency in log-log scaleis quite compatible with the logarithmic correctionsto a 1 / f 1/2 divergence predicted in reference [33].This power spectrum was computed from 23 samplesof 29 iterates of the mapping (6) using a Hammingwindow. Let us note, however, that the departure

Fig. 4. - The power spectrum as computed using theHamming windowing procedure from 23 samples of29 iterates of the Poincare map model (6) with the

parameter values given in equation (11) and

A = 1.709 x 10-4, v = 3.052 x 10-3 (In (e) = - 15). Thetime between two iterates of the mapping (6) has beennormalized to one second. In the insert, the powerspectrum is plotted versus In (f).

from the exponent d = 1/2 (which has been observedin the periodically driven oscillator (5) as well [16])can also be attributed to the fact that our numerical

experiment has been conducted at small but finitedistance from the intermittency threshold [13].

3. Type-11 intermittency in the presence of noise :numerical results.

When observations are compared with theory, it isessential to understand the role of external noise.

During the last few years, considerable progresshave been accomplished in understanding the effectof stochastic (as well as periodic) perturbations onsystems which exhibit a continuous transition to

chaos [35]. For type-I intermittency, the instabilityof the scenario [14, 35] to the presence of an externalnoise has been investigated numerically [2, 36] andtheoretically [37]. Renormalization group techniqueshave been used [3, 38-40] to predict the scalingbehavior of the order parameter, i. e. the envelope ofthe largest Lyapunov exponent (L ) or the inverse ofthe average length of the laminar episodes ( (n) - 1 ),as a function of the amplitude of the perturbation.To conclude this paper we extend this study to type-II intermittency. In fact we will limit ourselves to anumerical review of different situations correspond-ing to the insertion of additive and multiplicativenoises in the Poincare map model (6). Although wewill discuss only the numerical results obtained withthe three-dimensional discrete system (6), we havealso performed subsequent simulations of differentkinds of stochastic forcing of the two-dimensionalnormal form (1) of the subcritical Hopf bifurcationwith random uniformly distributed reentry points.Let us mention that these simulations corroboratethe scaling behaviors observed when iterating thePoincare map model (6) in the presence of noise.

3.1 ADDITIVE NOISE. - Figure 5a illustrates theeffect of additive random noise as modeled by theintroduction of the term hç in the third equation inthe Poincare map model (6) :

where C is a Gaussian random variable, with

C) = 0 and n’ C,,,) = Snn’ ; h is a variable thatcontrols the width (or amplitude) of the noise.Different values of 8 = (g - tk * ) have been investi-gated along the numerical path (11) ; the valuespresented in figure 5a correspond to In (e) = - 12,- 13 and - 14 respectively and belong to the rangeof e values where the inverse of the order parameter

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Fig. 5. - The average value n > of the laminar episodesversus the amplitude of the noise h in log-log scales. Themodel parameters are defined in equation (11). Severalvalues of e are considered : In (E) = - 12 (circles) ; - 13(triangles) ; - 14 (crosses). (a) Additive noise :

equation (15), the solid line corresponds to a linear

regression fit with an exponent a = - 1.19 ± 0.05 ; (b)Multiplicative noise : equation (16) ; (c) Multiplicativenoise : equation (17), the solid line is drawn to guide theeyes.

scales like (n) ’" E- 1/2 as predicted by equation (13)and shown in figure 3. Above some cut-off valueshc(E), the noise becomes more important than thedeterministic drift obtained beyond criticality andthe average length of the laminar episodes (n)decreases when one strengthens the noise. This

indicates that the intermittency threshold occurs for6 0, i.e. the transition to chaos is advanced in thepresence of additive noise. In a log-log plot, a linearregression fit yields an exponent a = - 1.19 ±

0.05, which is very likely to correspond to a linearincrease of the order parameter n > - 1 ’" L - h up tosome logarithmic corrections. These observations

clearly illustrate the sensitivity of type-II intermitten-cy to the presence of additive noise. The instabilityof this scenario is also characterized by a shift of thecross-over point which seems compatible with a

square-root power-law in s up to some logarithmiccorrections. Let us mention that we have obtainedthe same quantitative measurement when consider-ing the addition of random noise in the (subcritical)Hopf normal form (1).

3.2 MULTIPLICATIVE NOISE. - Figure 5b illustratesthe influence of a multiplicative noise as introducedthrough the coefficient kf = k1 + hg of the nonlinearterm x2 in the Poincare map model (6)

We consider values of h such that hx 2 covers thesame range of values as the amplitude (h ) of theadditive noise in figure 5a. For the three values of Epreviously investigated, (n) does not seem to beaffected (at least at leading order) by this particularmultiplicative noise ; subsequent computationsstrongly suggest that type-11 intermittency is stablewith respect to the presence of fluctuations in the

coefficients ki of the nonlinear terms in the three-dimensional mapping (6). This statement is furthercorroborated by direct simulations of the Hopfnormal form (1) in the presence of stochastic fluctu-ations in the coefficient a of the cubic nonlinear termin the modulus equation.

Figure 5c shows the case where the noise is

assigned to the control parameter ii = u + hç :

Unlike additive noise, beyond some cut-off valuehe (ê), the noise begins to affect the dynamics but insuch a way that (n) increases with h. This effect,which is also encountered when investigating theHopf normal form (1) with a noisy control parameterE, suggests that this time, the intermittencythreshold is very likely to be postponed (£ =/-t - IU * :. 0) by the noise. Unfortunately, because

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in this case the study of the statistics of the laminarperiods becomes increasingly time consuming, wehave not been able to extract the power law behavior

(if any) of n > as a function of the amplitude of thenoise.

This numerical analysis is only the prelude to atheoretical understanding of the effect of a randomnoise on type-11 intermittency. The reduction of thePoincare map model (6) in the presence of noise to anoisy Hopf normal form using stochastic center-

manifold and normal form techniques [41, 42] wouldbe a major step toward this goal. However, thisreduction would require to adapt these perturbativetechniques to discrete noisy systems which, to ourknowledge, is not yet established in the literature.Then with further approximation of this stochastictwo-dimensional map by a Focker-Planck equation[2, 37, 43-45], a quantitative analysis should be

attainable. As a first theoretical attempt [46], we

have used the axial symmetry of the Hopf normalform (1) to reduce this study to the analysis of a one-dimensional Focker-Planck equation. Within such aworking hypothesis, we have been able to under-stand some of our numerical results [46] e.g. the

exponent a = -1 extracted from the iteration ofthe discrete system (15). However, this study is stillpreliminary and incomplete since it does not takeinto account corrections which are very likely to beinduced by some breaking of the axial symmetry asobserved in our simulations. We hope to elaborateon this point in a forthcoming communication.

Acknowledgments.

We are very grateful to H. P. Herzel for veryinteresting discussions and to L. Tuckerman for acareful reading of the manuscript.

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[31] Note that with this one-dimensional reinjection pro-cess, the distribution of laminar episodes for

type-II intermittency becomes similar to the

distribution for type-III intermittency.

775

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[45] HORSTHEMKE, W. and LEFEVER, R., Noise-InducedTransitions : Theory and Applications in Physics,Chemistry and Biology, Springer Ser. in Synerge-tics (Springer, Berlin) 15 (1984) Chapter 4.

[46] ARGOUL, F., ARNEODO, A. and HERTZEL, H. P., inpreparation.